and even a commutative RR algebra (since RR is assumed to be commutative ring).

Similarly, for S=(S•):Δop→SetS = (S_\bullet) : \Delta^{op} \to Set a simplicial set write [S•,R][S_\bullet,R] for the cosimplicial algebra obtained by taking RR-valued functions in each degree. This is naturally

Theorem

In both these cases the complex of binary operations in these operads has a 0-cycle whose action C•(S,R)⊗C•(S,R)→C•(S,R)C^\bullet(S,R) \otimes C^\bullet(S,R) \to C^\bullet(S,R) is the usual cup product.

Proof

The statement for the Eilenberg–Zilber operad goes back to HinSch87 . A good review is in (May03) . The statement for the Barrat–Eccles operad is in (BerFre01) .